Global Existence Of Solutions To A Hyperbolic-Parabolic System

In biology, it is very important to investigate the movement of some cells or organisms in some given biological system. The mechanism of communication between cells or organisms depends on the different ways they interact. In many biological systems, the movement occurs in response to a diffusible substance or otherwise transported signal. Other systems are modeled by so-called short-range interactions due to local modifications of the environment such as the production and release of nutrients. In this case, dispersal is not simply one of simple diffusion but rather one of correlated or reinforced random walks.In this paper, we investigate the global existence of solutions to a hyperbolic-parabolic model of chemotaxis arising in the theory of reinforced random walks. To get L²-estimates of solutions, we construct a nonnegative convex entropy of the corresponding hyperbolic system. The higher energy estimates are obtained by the energy method and a priori assumptions.The main result of this paper is:Theorem 2.2. Let u₀－1∈H²([0,1]) and ||u₀||₂²+||u₀－1||₂² be sufficiently small. Then there exists a unique global solution (u(x,t),v(x,t)) of (2.1)-(2.3), satisfies:...